Add the following rational expressions. $\dfrac{2z^2-3z}{z^2+1}+\dfrac{4z+2}{z^2+1}=$
We want to add two rational expressions whose denominators are equal. We can do this by adding the numerators and keeping the denominator the same. [Does this fit with how we add rational numbers?] $\begin{aligned} &\phantom{=}\dfrac{2z^2-3z}{z^2+1}+\dfrac{4z+2}{z^2+1} \\\\ &=\dfrac{(2z^2-3z)+(4z+2)}{z^2+1} \\\\ &=\dfrac{2z^2-3z+4z+2}{z^2+1} \\\\ &=\dfrac{2z^2+z+2}{z^2+1} \end{aligned}$ In conclusion, $\dfrac{2z^2-3z}{z^2+1}+\dfrac{4z+2}{z^2+1}=\dfrac{2z^2+z+2}{z^2+1}$